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A119441
Distribution of A063834 in Abramowitz and Stegun order.
5
1, 2, 1, 3, 2, 1, 5, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 11, 7, 10, 9, 5, 6, 8, 3, 4, 2, 1, 15, 11, 14, 15, 7, 10, 9, 12, 5, 6, 8, 3, 4, 2, 1, 22, 15, 22, 21, 25, 11, 14, 15, 20, 18, 7, 10, 9, 12, 16, 5, 6, 8, 3, 4, 2, 1, 30, 22, 30, 33, 35, 15, 22, 21
OFFSET
1,2
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
T(n,k) = product_{p=1..A036043(n,k)} A000041(c), 1<=k<=A000041(n), where c are the parts in the k-th partition of n. - R. J. Mathar, Jul 12 2013
EXAMPLE
1;
2, 1;
3, 2, 1;
5, 3, 4, 2, 1;
7, 5, 6, 3, 4, 2, 1;
T(5,2) = 5 because the second partition of 5 is 1+4 and 4 can be repartitioned in 5 different ways.
T(5,3) = 6 because the third partition of 5 is 2+3, where the 2 can be partitioned in 2 ways (2, 1+1) and the 3 can be partitioned in 3 ways (3, 1+2, 1+1+1), 6=2*3.
T(5,4) = 3 because the fourth partition of 5 is 1+1+3 and 3 can be partitioned in 3 different ways.
MAPLE
# Compare two partitions (list) in AS order.
AScompare := proc(p1, p2)
if nops(p1) > nops(p2) then
return 1;
elif nops(p1) < nops(p2) then
return -1;
else
for i from 1 to nops(p1) do
if op(i, p1) > op(i, p2) then
return 1;
elif op(i, p1) < op(i, p2) then
return -1;
end if;
end do:
return 0 ;
end if;
end proc:
# Produce list of partitions in AS order
ASPrts := proc(n)
local pi, insrt, p, ex ;
pi := [] ;
for p in combinat[partition](n) do
insrt := 0 ;
for ex from 1 to nops(pi) do
if AScompare(p, op(ex, pi)) > 0 then
insrt := ex ;
end if;
end do:
if nops(pi) = 0 then
pi := [p] ;
elif insrt = 0 then
pi := [p, op(pi)] ;
elif insrt = nops(pi) then
pi := [op(pi), p] ;
else
pi := [op(1..insrt, pi), p, op(insrt+1..nops(pi), pi)] ;
end if;
end do:
return pi ;
end proc:
A119441 := proc(n, k)
local pi, a, p ;
pi := ASPrts(n)[k] ;
a := 1 ;
for p in pi do
a := a*combinat[numbpart](p) ;
end do:
a ;
end proc:
for n from 1 to 10 do
for k from 1 to A000041(n) do
printf("%d, ", A119441(n, k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, Jul 12 2013
CROSSREFS
Cf. A063834, A119442, A000041 (row lengths and also first column)
Sequence in context: A304100 A179314 A204927 * A347227 A322083 A058399
KEYWORD
easy,nonn,tabf
AUTHOR
Alford Arnold, May 19 2006
STATUS
approved